Antiferromagnetic magnons as highly squeezed Fock states underlying quantum correlations Akashdeep Kamra,1, ∗ Even Thingstad,1 Gianluca Rastelli,2, 3 Rembert A. Duine,4, 5, 1 Arne Brataas,1 Wolfgang Belzig,2 and Asle Sudbø1 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway 2Department of Physics, University of Konstanz, Konstanz, Germany 3Zukunftskolleg, University of Konstanz, Konstanz, Germany 4Utrecht University, Utrecht, The Netherlands 5Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands Employing the concept of two-mode squeezed states from quantum optics, we demonstrate a re- vealing physical picture for the antiferromagnetic ground state and excitations. Superimposed on a N´eelordered configuration, a spin-flip restricted to one of the sublattices is called a sublattice- magnon. We show that an antiferromagnetic spin-up magnon is comprised by a quantum superposi- tion of states with n+1 spin-up and n spin-down sublattice-magnons, and is thus an enormous excita- tion despite its unit net spin. Consequently, its large sublattice-spin can amplify its coupling to other excitations. Employing von Neumann entropy as a measure, we show that the antiferromagnetic eigenmodes manifest a high degree of entanglement between the two sublattices, thereby establishing antiferromagnets as reservoirs for strong quantum correlations. Based on these insights, we outline strategies for exploiting the strong quantum character of antiferromagetic (squeezed-)magnons and give an intuitive explanation for recent experimental and theoretical findings in antiferromagnetic magnon spintronics. I. INTRODUCTION teractions, such as dipolar fields or crystalline anisotropy, underlie the magnon-squeezing in ferromagnets. These As per the Heisenberg uncertainty principle, the quan- spin-nonconserving interactions were further found to re- tum fluctuations of two non-commuting observables can- sult in two-sublattice magnets hosting excitations with not simultaneously be reduced to zero. However, it is spin varying continuously between positive and nega- 20 possible to generate a state with the quantum noise in one tive values . In contrast, exchange interaction in a observable reduced below its ground state limit at the ex- two-sublattice magnet leads to a strong squeezing ef- pense of enhanced fluctuations in the other observable1,2. fect, which does not affect the excitation spin and forms Considering a single mode or frequency of light, such a main subject of the present article. Being eigen- states, generally called squeezed vacuum1,2, have proven excitations, squeezed-magnons are qualitatively distinct instrumental in the detection of gravitational waves3 with in certain ways from the squeezed states of light discussed a sensitivity beyond the quantum ground state limit4{6. above, which are non-equilibrium states generated via an Furthermore, squeezed vacuum states have applications external drive. At the same time, the two kinds of states in quantum information7{11 since they exhibit quantum share several similar features on account of their wave- correlations and entanglement. These are best repre- functions being mathematically related. To emphasize sented and exploited via the two-mode squeezed vac- this difference, we employ the terminology that \squeezed uum states, where the two participating modes are en- state of a boson" refers to a non-equilibrium state, while 23 tangled and correlated1. The widely studied1,2 single- a \squeezed-boson" is an eigenmode . and two-mode squeezed vacuums may be considered a Instigated by recent experimental breakthroughs24{29, special case, corresponding to zero photon number(s), of interest in antiferromagnets (AFMs) for practical appli- a wider class - squeezed Fock states12,13. While investi- cations has been invigorated30{34. Due to the well-known gated theoretically, the latter have been largely forgotten, strong quantum fluctuations in AFMs, they have also probably owing to the experimental challenge of gener- been the primary workhorse of the quantum magnetism ating them. The squeezing concept applies to bosonic community35. The N´eelordered configuration, which is arXiv:1904.04553v3 [cond-mat.mes-hall] 5 Nov 2019 modes in general, and squeezed states of magnons14{17 consistent with most of the experiments, is not the true and phonons18 have also been achieved experimentally. quantum ground state of an AFM. Furthermore, quan- The concept of squeezed Fock states12,13 has proven tum fluctuations destroy any order in a one-dimensional valuable in understanding the spin excitations of or- isotropic AFM. These and related general ideas applied dered magnets19,20. Squeezed-magnons have been shown to AFMs bearing geometrically frustrated interactions to be the eigen-excitations of a ferromagnet19,21.A underlie quantum spin liquids36{38, which are devoid of squeezed-magnon is comprised by a coherent superpo- order in the ground state and host exotic, topologically sition of the different odd number states of the spin-1 non-trivial excitations embodying massive entanglement. magnon19,2022. This bestows it a noninteger average spin We here develop the squeezing picture for the ground larger than 1. The relatively weak spin-nonconserving in- state and excitations of a simple, two-sublattice AFM. It 2 continuously connects and allows a unified understand- ing of classical and quantum as well as ordered and disor- dered antiferromagnetic states. We show that the AFM eigenmodes are obtained by pairwise, two-mode squeez- (a) ing of sublattice-magnons, the spin-1 excitations delocal- ized over one of the two sublattices. Focusing on spatially uniform modes, the antiferromagnetic ground state is a superposition of states with equal number of spin-up and -down sublattice-magnons [Fig. 1(a) and (c)]. The result is a diminished net spin on each sublattice by an amount (b) dictated by the degree of squeezing, parametrized by the non-negative squeeze parameter r. Similarly, a spin- up AFM (squeezed-)magnon is comprised by a super- position of states with n + 1 spin-up and n spin-down sublattice-magnons [Fig. 1(b) and (c)]. Thus, despite its unit net spin, it carries enormous spins on each sublattice which allows it to couple strongly with other excitations via a sublattice-spin mediated interaction (Fig. 2). Ow- ing to a perfect correlation between the two sublattice- magnon numbers, AFM squeezed-magnons are shown to embody entanglement quantified by von Neumann en- tropy1,39 increasing monotonically with r (Fig. 3). The degree of squeezing and entanglement embodied by these eigenmodes is significantly larger than that in hitherto achieved non-equilibrium states. We also comment on existing experiments40,41, where this squeezing-mediated (c) coupling enhancement (Fig. 2) has been observed, and FIG. 1. Schematic depiction of spatially uniform antifer- strategies for exploiting the entanglement contained in romagnetic (a) vacuum and (b) spin-up eigenmodes. (a) antiferromagnetic magnons. While the squeezed states of P The vacuum mode, represented as j0isq = n Pn jn; nisub, light are generated via external drives and are nonequi- is a superposition over states with equal number of spin-up librium states1, the antiferromagnetic squeezed-magnons and -down sublattice-magnons. (b) The spin-up squeezed- P are eigenmodes of the system with their squeezing being magnon, represented as j"isq = n Qn jn + 1; nisub, is com- equilibrium in nature and resulting from energy mini- prised by states with one extra spin-up sublattice-magnon. (c) mization. Squared amplitudes corresponding to the sublattice-magnon states constituting the uniform squeezed vacuum and spin-up eigenmodes for squeeze parameters of 3 (main) and 1 (inset). II. AFM EIGENMODES AS SQUEEZED FOCK scribed by: STATES J X ~ ~ H~uni = S A(ri) · S B(ri + δ) ~2 We consider a N´eelordered ansatz with sublattice A i;δδ z^ z^ K X h i2 K X h i2 and B spins pointing along z and −z, respectively. The − S~ (r ) − S~ (r ) : (2) antiferromagnetic Hamiltonian may then be expressed in 2 Az i 2 Bz j ~ i ~ j terms of the corresponding sublattice-magnon ladder op- ~ 20,42 eratorsa ~k ; bk as : Here, the positive parameters J and K account for in- tersublattice antiferromagnetic exchange and easy-axis ~ anisotropy, respectively. S A;B represent the respective ~ X y ~y~ ~ y~y H = Ak a~k a~k + bk bk + Ck a~k b−k +a ~k b−k ; (1) spin operators, ri (rj) runs over the sublattice A (B), k and δ are vectors to the nearest neighbors. Executing Holstein-Primakoff transformations43 and switching to Fourier space, Eq. (2) reduces to Eq. (1) apart from a where we assume inversion symmetry and disregard ap- constant energy offset20,44, with A = JSz + 2KS and plied magnetic fields, for simplicity. Consistent with the k Ck = JSzγk . Here, S is the spin on each site, z is the assumed N´eel order, sublattice B (A) magnons repre- P coordination number, and γk ≡ (1=z) exp (ikk · δ). sented by ~b (~a ) are spin-up (-down). In addition to δ k k The Hamiltonian [Eq. (1)] is diagonalized to H~ = the general considerations captured by Eq. (1), we will P y ~y ~ 43 obtain specific results for a uniaxial, easy-axis AFM de- k k α~k α~k + βk βk via a Bogoliubov transformation 3 described by45: ~y ~ ~ y α~k = uk a~k + vk b−k ; βk = uk bk + vk a~−k ; (3) r r Ak + k Ak − k uk = ; vk = ; (4) 2k 2k q 2 2 ~ where k = Ak − Ck .α ~k and βk represent the spin- down and -up eigenmodes of the AFM, which are subse- (a) (b) quently called squeezed-magnons. Denoting the resulting antiferromagnetic vacuum or ground state wavefunction FIG. 2. (a) An external excitation bath (shaded green) inter- ~ acts weakly with the AFM squeezed-magnon if coupled via by jGisq, we haveα ~k jGisq = βk jGisq = 0 for all k. Let us first consider the spatially uniform modes, i.e. its unit net spin (left), but strongly if exposed to only one of the sublattices (right).